Charles Babbage's Difference Engine is one of the great computing devices that got away. First designed in the 1840s as a massive automated calculator for the Victorian era, Babbage was unable to actually build the thing in his lifetime, mainly due to his limited budget and its tremendous size. (Note: Ada Lovelace wrote programs for Babbage's more advanced Analytical Engine, also unbuilt in his lifetime.) After working through various computing machine designs, Babbage arrived at the Difference Engine No. 2 design, reducing the number of parts required and improving the scope of calculations. He didn't build that one either.
Modern engineers constructed the Difference Engine No. 2 using Victorian era technology and proved that it worked, after sorting out some minor bugs. Here's a modern demo, including an extensive history lesson on Babbage, at the Computer History Museum:
If you'd rather take the super-short version, here's a history lesson (minus the detailed in-person demo, trivia, etc.):
For more, check out The Computer History Museum's mini-site about the Babbage engines.